Fall, 2001

Monday & Friday, 8:00-8:50 a.m

Tuesday & Thursday, 8:25-9:15 a.m.

Buttrick G-13

Office Hours. Monday, 9:00-10:00, Thursday, 10:00-11:00. If these hours are not good for you, we can set up an appointment.

Prerequisites. Math 204 (The Art Of Mathematical Thinking) and Math 206 (Linear Algebra).

Text. A First Course in Abstract Algebra, 6^th edition, by John B. Fraleigh.

Course Content. Chapters 1-7 of the text will be studied. Topics include groups, factor groups, cyclic groups, rings, prime and maximal ideals, fields, homomorphisms and isomorphisms, and factorization.

What is "Abstract Algebra"? Well, first we ask, "what is algebra?" Simply put, algebra is about solving equations. In high school algebra, all of the equations have real variables, i.e. variables whose values are assumed to be real numbers. This is rather restrictive, for there are many other very nice number systems out there. In abstract algebra, we try to solve equations where the variables come from different number systems. Much of this class consists of asking, "what other number systems are out there?"

What good is "Abstract Algebra"? At first, it may seem like this class is only an exercise in abstract mathematical thinking. However, there are many applications to what we will be studying. Abstract algebra plays a vital role in:

• Automata: the development of finite-state machines, for example telephones and vending machines.
• Business and Economics: maximizing a set of linear equations constrained by a set of linear inequalities.
• Computer Graphics: 3-D rotations, movement of figures, etc.
• Cryptography: creating and breaking secret codes.
• Information Theory: the design of efficient error-correcting codes.
• Physics: the state of sub-atomic particles.
• Quantum Chemistry: the movements of electrons.

Course Goals. By the end of the semester, you should:

• Understand the notion of abstract algebra

• Be able to write a concise algebra proof

• Be able to read and understand technical mathematics

Class Description. With the exception of the days mentioned below, this will be conducted as a lecture-style class. That is not to say your input won’t be valued or solicited: frequently questions will be posed of you, and (hopefully) questions will be posed of me. If it any time you have a question, please feel free to speak up rather than raise your hand. That’s a power trip I just don’t need.

"Q&A". Six days are designated "Q&A". On these days, we will have time for you to both ask and answer questions. We will take as many questions as possible, and someone will have the opportunity to answer the questions at the board. Of course, you will be rewarded for providing solutions. If you don’t believe me, see "Participation" below.

Attendance. You are expected to attend every class. While attendance during exams and Thursday Q&A will directly affect your grade, missing the other days does not have a direct impact on your grade (except during oral presentations). It will, however, have an indirect impact on your grade. Trust me. Also, you will not be penalized directly for tardiness, however you are expected to arrive to each class on time. Oh, and by the way…you are paying $40.51 for each lecture.

ecademy.agnesscott.edu. A web page has been set up for this course. You can find it at http://ecademy.agnesscott.edu/Mathematics/mathfacpgs/koch,alan/index.htm, or if that’s too much typing you can just follow the links from the http://ecademy.agnesscott.edu page. Here you will find all the handouts for the course. Most will be .pdf files, so make sure the computer you’re using has Adobe Acrobat installed.

Here’s how you’ll be graded…

Homework. Each night, there will be homework problems assigned from the sections covered during the lecture. They will consist (primarily) of odd problems from the book, so you can check your answers in the back. This homework will not be collected, but it is assumed that it will be completed by the start of the next class. (Obviously, you won’t be graded on this, but it seemed the most logical place to put this information.)

Participation. You will be graded on participation for this course. Factors that will be used in computation of "participation" include both the asking and answering of questions in class, as well as your attendance during your classmates’ presentations. For Q&A, providing solutions to questions will surely help your grade.

Assignments. There will be six assignments that you will turn in. While the questions will tend to be more theoretical than the problems assigned after each section, it is important that you do the daily homework to get a feel for what is going on before you attempt the assignments. Assignments are due by the start of class on the day indicated at the top of the problem set. Late assignments will not be accepted.

You are encouraged to discuss these assignments with the others in the class, but your write-up must be your own. If you have any question about this policy, please let me know.

Honor Code. All students are expected to follow the honor code throughout the semester. Any graded work, be it an assignment or an exam, must be pledged (and signed) in order for it to be graded. Please consult the student handbook for more details.

Exams. You will have three take home exams. The exams will tend to focus more on the calculation-type problems than the theoretical, although this should not be taken as a guarantee. These are tentatively scheduled to be given September 24, November 1, and December 3. If you have a conflict with any of these dates, let me know ASAP. (The day after the exam is not ASAP.)

The exams will cover material from the text, along with material presented in class. You will not be allowed to make-up an exam without a doctor’s note.

Final Exam. The final exam is cumulative.

Oral Presentation. In addition to the assignments listed above, you must also complete an oral presentation. The purpose of this presentation is to

1. give you more exposure to an application of abstract algebra

2. provide you with an easy forum to read a scientific document

3. give you practice in formal mathematical lecturing

The presentations will take place during the last four days of class.

Once a topic is picked, you are to prepare a 45-minute talk on the subject. You are to teach your topic to the rest of the class at a level consistent with the level of someone who has just successfully completed abstract algebra. It is helpful to provide basic definitions, give examples, etc. The four topics are as follows:

• Binary Linear Codes

• Cayley Digraphs

• Elliptic Curves

• Representation Theory

You will (hopefully) not have enough time in 45 minutes to completely describe your topic. Be prepared to construct a talk that lasts 45 minutes. Make sure it has a definite conclusion – do not just keep talking and talking until your 45 minutes are up. Be prepared to take questions from the class (including from me). For the latter two topics, you will need to borrow a book from me (or get the book from another library – ASC does not carry good titles in elliptic curves or in representation theory).

You should not expect a curve to be applied to the point scale, although plusses and minuses will be added as appropriate.